Lemma 20.25.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ X$-modules. If $H^ i(U_{i_0 \ldots i_ p}, \mathcal{F}^ q) = 0$ for all $i > 0$ and all $p, i_0, \ldots , i_ p, q$, then the map $ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \to R\Gamma (X, \mathcal{F}^\bullet ) $ of Lemma 20.25.1 is an isomorphism.

**Proof.**
Immediate from the spectral sequence of Lemma 20.25.1.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: